📚 Vectors in the Plane: Introduction and Operations | 平面向量：基础与运算
Welcome to Lesson 16-1 of the Student’s Edition for IGCSE Mathematics. In this lesson we meet vectors, quantities that carry both a size and a direction. Vectors appear in the IGCSE syllabus in topics such as transformations, kinematics, and geometry, so a secure understanding of the basics will serve you well across the whole paper.
欢迎来到 IGCSE 数学学生版第 16 单元第 1 课。本课我们认识“向量”——一种同时具有大小和方向的量。向量在 IGCSE 考纲中出现在变换、运动学与几何等章节中，因此扎实掌握基础概念对你完成整份试卷都大有帮助。
1. What Is a Vector? | 什么是向量？
A vector is a quantity that has both magnitude (size) and direction. Displacement, velocity, and force are everyday examples. A quantity that has only magnitude, such as distance, speed, or mass, is called a scalar.
向量是一种同时具有“大小（模）”和“方向”的量。位移、速度、力都是日常生活中的向量例子。而只有大小、没有方向的量，如距离、速率、质量，称为“标量”。
For instance, walking 5 km north describes a displacement vector: the 5 km is the magnitude, and “north” tells us the direction. Simply saying “5 km” leaves the direction unknown.
例如，“向北走 5 千米”描述了一个位移向量：其中 5 千米是大小，“北”是方向。若只说“5 千米”，则方向不明。
2. How Do We Write a Vector? | 向量的表示方法
In geometry, you will often see a vector written with an arrow: AB⃗ (read as “vector AB”) means the displacement from point A to point B. In print, vectors are also shown by bold letters such as v, or by a single letter with an arrow above it: v⃗.
在几何中，我们常用带箭头的记号表示向量：AB⃗（读作“向量 AB”）表示从点 A 指向点 B 的位移。在印刷体中，向量也常用粗体字母表示，如 v，或在字母上方加箭头：v⃗。
On the coordinate plane, a vector can be described by its horizontal and vertical components. A vector that moves 3 units right and 4 units up can be written as a column vector. In your exam, you will stack the two numbers vertically; in this article we write it as (3, 4), meaning 3 horizontally and 4 vertically.
在坐标平面中，向量可用它的水平分量与竖直分量来描述。一个“向右 3 个单位、向上 4 个单位”的向量可以写成列向量。考试中，你需要把两个数上下排列；为了方便网页排版，本文写作 (3, 4)，表示水平方向 3，竖直方向 4。
Both notations describe exactly the same vector. As long as the two components are known, the ray’s length and direction are fully determined.
这两种写法描述的是同一个向量。只要两个分量已知，该有向线段的长短和方向就完全确定了。
3. The Magnitude of a Vector | 向量的模（长度）
The magnitude of a vector is its length. For a vector with components (x, y), we use Pythagoras’ theorem to find its magnitude:
向量的模就是它的长度。对于分量为 (x, y) 的向量，我们用勾股定理求其模：
|v⃗| = √(x² + y²)
Example: the vector (3, 4) has magnitude √(3² + 4²) = √25 = 5. Notice that the vector (6, 8) also has magnitude √(36 + 64) = √100 = 10, which is exactly twice the original: (6, 8) = 2 × (3, 4).
例如：向量 (3, 4) 的模为 √(3² + 4²) = √25 = 5。注意向量 (6, 8) 的模为 √(36 + 64) = √100 = 10，恰好是原来的两倍：(6, 8) = 2 × (3, 4)。
Magnitude is always a positive scalar (or zero, for the zero vector). It has no direction attached to it.
模永远是一个正标量（零向量则为 0）。它本身不包含方向信息。
4. Adding and Subtracting Vectors | 向量的加法与减法
To add two vectors, you may draw them “tip to tail”: place the start of the second vector at the tip of the first; the resultant goes from the tail of the first to the tip of the second.
向量相加时，可以用“首尾相接”的方法作图：把第二个向量的起点放在第一个向量的终点，从第一个向量的起点到第二个向量的终点连成的向量就是它们的和。
Component-wise, we simply add the corresponding components:
在分量层面，我们只需把对应分量相加：
(a, b) + (c, d) = (a + c, b + d)
Example: (2, 3) + (5, -1) = (7, 2). Similarly, subtraction is done component-wise: (a, b) − (c, d) = (a − c, b − d). Geometrically, subtracting means adding the opposite (negative) vector.
例如：(2, 3) + (5, −1) = (7, 2)。同样地，减法也是逐分量处理：(a, b) − (c, d) = (a − c, b − d)。从几何上看，减去一个向量相当于加上它的相反向量。
Always check whether the result is reasonable by a quick sketch; a diagram helps you avoid sign errors.
务必用简图检查结果是否合理，画图能帮你避免符号错误。
5. Multiplying by a Scalar | 向量的数乘
When a vector is multiplied by a positive scalar k, its length is multiplied by k and its direction stays the same. When k is negative, the direction reverses.
当向量乘以正标量 k 时，其长度变为原来的 k 倍，方向不变；当 k 为负数时，方向反转。
k × (x, y) = (kx, ky)
For example, 2 × (3, 4) = (6, 8). The vector −1 × (3, 4) = (−3, −4) has the same magnitude as (3, 4) but points in the opposite direction.
例如，2 × (3, 4) = (6, 8)。而 −1 × (3, 4) = (−3, −4) 的模与 (3, 4) 相同，但方向相反。
This operation is called scalar multiplication. It is essential in questions about parallel vectors and in vector geometry.
这种运算叫作“数乘”。在平行向量与向量几何问题中它至关重要。
6. Parallel and Equal Vectors | 平行向量与相等向量
Two vectors are parallel if one is a scalar multiple of the other. For example, (2, 5) and (6, 15) are parallel because (6, 15) = 3 × (2, 5).
若一个向量是另一个向量的标量倍，则称这两个向量平行。例如，(2, 5) 与 (6, 15) 平行，因为 (6, 15) = 3 × (2, 5)。
Two vectors are equal if they have the same magnitude and the same direction. Equal vectors are also parallel, but parallel vectors are not necessarily equal (they may have different lengths or point in opposite directions).
两个向量相等，当且仅当它们的大小和方向都相同。相等向量一定平行，但平行向量不一定相等（它们的长度可以不同，或方向相反）。
In a diagram, arrows of the same length and direction represent the same vector even if they are drawn in different positions. A vector does not have a fixed starting point; only its displacement matters.
在图形中，长度和方向相同的箭头即使画在不同位置，也代表同一个向量。向量没有固定的起点，只有位移本身有意义。
7. Position Vectors | 位置向量
A position vector describes a point relative to the origin. If point P has coordinates (x, y), then the position vector of P, written as p⃗ or OP⃗, is simply (x, y). The arrow always starts at O = (0, 0).
位置向量描述的是“相对于原点”的某一点。若点 P 的坐标为 (x, y)，则 P 的位置向量记为 p⃗ 或 OP⃗，其分量就是 (x, y)，箭头始终从原点 O = (0, 0) 出发。
Position vectors are extremely useful because they connect coordinates with vector algebra. For example, if A = (1, 2) and B = (5, 9), we can represent them as a⃗ = (1, 2) and b⃗ = (5, 9).
位置向量非常有用，因为它把坐标与向量代数联系起来。例如，若 A = (1, 2)，B = (5, 9)，则 a⃗ = (1, 2)，b⃗ = (5, 9)。
Remember: a free vector can slide anywhere, but a position vector is pinned to the origin.
请记住：自由向量可以任意平移，但位置向量被“钉”在原点上。
8. The Vector between Two Points | 两点间的向量
One of the most important formulas in this lesson: the vector from A to B is obtained by subtracting the position vector of A from that of B.
本课最重要的公式之一：从 A 指向 B 的向量，等于用 B 的位置向量减去 A 的位置向量。
AB⃗ = b⃗ − a⃗ = (x_B − x_A, y_B − y_A)
Example: if A = (1, 2) and B = (5, 9), then AB⃗ = (5 − 1, 9 − 2) = (4, 7). Check by drawing: starting at A, move 4 right and 7 up to reach B.
例如：若 A = (1, 2)，B = (5, 9)，则 AB⃗ = (5 − 1, 9 − 2) = (4, 7)。你可以画图验证：从 A 出发，向右 4、向上 7 即到达 B。
Note the order matters: BA⃗ = a⃗ − b⃗ = (−4, −7). The vectors AB⃗ and BA⃗ are negatives of each other.
注意顺序很重要：BA⃗ = a⃗ − b⃗ = (−4, −7)。向量 AB⃗ 与 BA⃗ 互为相反向量。
9. Vector Geometry and Ratio Problems | 向量几何与比例问题
Vectors can prove results about straight lines, midpoints, and points dividing segments. The midpoint M of AB has position vector (a⃗ + b⃗) ÷ 2, because M is exactly halfway along the segment.
向量可以用来证明关于直线、中点与分点的一些结论。AB 的中点 M 的位置向量为 (a⃗ + b⃗) ÷ 2，因为 M 恰好位于线段的一半处。
More generally, if point P divides AB in the ratio m : n (with AP : PB = m : n), then the position vector of P is
更一般地，若点 P 把线段 AB 分成 m : n（即 AP : PB = m : n），则 P 的位置向量为
p⃗ = (n a⃗ + m b⃗) ÷ (m + n)
For example, if A = (1, 2) and B = (4, 5), the point that divides AB in the ratio 2 : 1 has position vector (1 × (1, 2) + 2 × (4, 5)) ÷ 3 = (9, 12) ÷ 3 = (3, 4).
例如，若 A = (1, 2)，B = (4, 5)，且按 2 : 1 分割 AB 的点，其位置向量为 (1 × (1, 2) + 2 × (4, 5)) ÷ 3 = (9, 12) ÷ 3 = (3, 4)。
Geometry questions often ask you to express a vector in terms of two independent base vectors, then compare coefficients to find unknown lengths or ratios.
几何题经常会要求你用两个独立的基向量表示某个向量，然后通过比较系数求出未知长度或比例。
10. Common Pitfalls and Exam Tips | 常见错误与应试技巧
Below is a checklist of the mistakes students most often make in vector questions, together with advice to avoid them.
下面是学生在向量题中最常犯的错误清单，以及相应的避错建议。
- The order of subtraction in AB⃗ = b⃗ − a⃗ is often confused. Always ask “from where to where?” From A to B means start at A, so subtract a⃗.
- AB⃗ = b⃗ − a⃗ 中减法的顺序经常被搞混。永远问自己“从哪到哪？”从 A 到 B 表示从 A 出发，所以要减去 a⃗。
- Forgetting to show working when adding column vectors. Write each component addition explicitly in the exam.
- 列向量相加时不写过程。考试中要把每个分量的加法明确写出来。
- Confusing position vector with free vector. A position vector always starts at the origin; a free vector may start anywhere.
- 混淆位置向量与自由向量。位置向量总是从原点出发；自由向量可放在任意位置。
- Ignoring the scale of the diagram. Use the given grid or known lengths; never measure with a ruler unless the question says the diagram is accurate.
- 忽略图形比例。要使用题目给定的网格或已知长度；除非题目说明按实际比例绘制，否则不要用尺子去量。
Finally, always mark the direction of the arrow clearly on your diagrams. A vector without an arrow is just a length; the direction is part of its identity.
最后，在图中一定要清楚标出箭头方向。没有箭头的“向量”仅仅是一个长度；方向是向量身份的一部分。
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